Mock Geometry AIME 2011 Problems/Problem 3

Problem

In triangle $ABC,$ $BC=9.$ Points $P$ and $Q$ are located on $BC$ such that $BP=PQ=2,$ $QC=5.$ The circumcircle of $APQ$ cuts $AB,AC$ at $D,E$ respectively. If $BD=CE,$ then the ratio $\frac{AB}{AC}$ can be expressed in the form $\frac{m}{n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

$[asy] unitsize(1cm); draw((-1,0)--(8,0)--(0,4)--cycle); draw((0,4)--(1,0)); draw((0,4)--(3,0)); label("$A$",(0,4),N); label("$B$",(-1,0),WSW); label("$C$",(8,0),ESE); label("$P$",(1,0),S); label("$Q$",(3,0),S); draw(circumcircle((0,4),(1,0),(3,0))); label("$D$",(-0.6,2),SW); label("$E$",(4.5,1.7),NE); [/asy]$

By the Power of a Point Theorem on $B$ , we have $BD \times BA=BP \times BQ=2 \times 4=8$ . By the Power of a Point on $C$ , we have $CE \times CA=CQ \times CP=5 \times 7=35$ . Dividing these two results yields $\frac{BD \times BA}{CE \times CA}=\frac{8}{35}$ . We are given $BD=CE$ and so $\frac{BD}{CE}=1$ . Then the previous equation simplifies to $\frac{AB}{AC}=\frac{8}{35}$ . Hence $m+n=8+35=\boxed{043}$

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Mock Geometry AIME 2011 Problems/Problem 3

Problem

Solution